The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of spaces $S^{m,k}$ where we defines $$ S^{m,k} := SU(m+1)/(SU(k) \times SU(m-k)), ~~~~~~~~~~~~~~ k = 1, \ldots, m - 1. $$ It also have a $U(1)$-action, and quotienting by this give the Grassmanian $Gras(m,k)$.

What is correct the name and notation for $S^{m,k}$?

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    $\begingroup$ "The unit vectors in $\mathcal O_{Gr(m,k)}(-1)$", or wordlessly, perhaps $\mathcal O_{Gr(m,k)}(-1)_{||^2=1}$. $\endgroup$ – Allen Knutson Jun 22 '16 at 17:42
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    $\begingroup$ "Canonical circle bundle over the Grassmannian"? $\endgroup$ – paul garrett Jun 22 '16 at 19:09
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    $\begingroup$ "The unitary frame bundle of the tautological line bundle of the Grassmannian." $\endgroup$ – Andreas Cap Jun 23 '16 at 6:50

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